Is it permissible for a woman who is عرفيًا (customarily/informally) married to marry another man if he proposes to her?

Marriage has pillars, namely: the woman's guardian, two just witnesses, and a formula indicating the contract. If some of these pillars are missing, the marriage is invalid. If all of them are missing, then cohabitation is considered blatant adultery, which is a major sin. What was mentioned about the notoriety of the aforementioned marriage, the woman's declaration that you are her husband, and your spending on her does not substitute for a legitimate contract. It is obligatory to distance yourself from her immediately and repent to Allah. You are permitted to contract a legitimate marriage with her, with its pillars, after repentance and after she has cleansed herself with one menstrual cycle. Similarly, she is permitted to marry any suitor after cleansing herself with one menstrual cycle.